mathematics

Article

Analysis of Queueing System MMPP/M/K/K withDelayed Feedback

Agassi Melikov 1, Sevinj Aliyeva 2 and Janos Sztrik 3,*1 Institute of Control Systems, National Academy of Science, Baku AZ1141, Azerbaijan;

agassi.melikov@gmail.com2 Information Technology and Programming, Applied Mathematics and Cybernetics, Baku State University,

Baku AZ1141, Azerbaijan; saliyeva@bsu.edu.az3 Department of Informatics and Networks, Faculty of Informatics, University of Debrecen, Debrecen 4032,

Hungary* Correspondence: sztrik.janos@inf.unideb.hu

Received: 22 October 2019; Accepted: 15 November 2019; Published: 18 November 2019 �����������������

Abstract: The model of multi-channel queuing system with Markov modulated Poisson process(MMPP) flow and delayed feedback is considered. After the customer is served completely, they willdecide either to join the retrial group again for another service (feedback) with some state-dependentprobability or to leave the system forever with complimentary probability. Feedback calls organizean orbit of repeated calls (r-calls). If upon arrival of an r-call all the channels of the system are busy,then it either leaves the system with some state-dependent probability or with a complementaryprobability returns to orbit. Methods to calculate the steady-state probabilities of the appropriatethree-dimensional Markov chain as well as performance measures of investigated system aredeveloped. Results of numerical experiments are demonstrated.

Keywords: multi-channel queuing system; delayed feedback; finite and infinite orbit; calculationmethods

1. Introduction

Most of the papers on queuing theory have considered the systems without feedback, i.e. withoutre-service phenomena. However, queues with feedback occur in many practical situations, e.g.,multiple access telecommunication systems, where data that is transmitted erroneously are sent againcan be modeled as queue with feedback.

We will distinguish between re-service of two types: instantaneous and delayed re-service. In thecase of instantaneous repetition, after completion of the previous service, the customer either requiresre-service immediately with some positive probability or leaves the system forever with complimentaryprobability. In the case of delayed re-service, after the customer is served completely, they will decideeither to join the retrial group again for another service with some positive probability or to leave thesystem forever with complimentary probability. Such kinds of feedbacks are called Bernoulli feedback.

Pioneer works related to queuing systems with feedbacks of both types were Takacs’s papers [1,2].Note that the model of the queuing system with instantaneous feedback (IFB) was investigated inReference [1], while in Reference [2] the model of the queuing system with delayed feedback (DFB)was considered. In both papers, the method of generating functions was used.

In the last three decades, various authors have intensively studied models of queuing systemswith feedback of both types. In this paper, we investigated the model of the queuing system with DFB.A detailed review of a large number of publications in that direction can be found in Reference [3].Therefore, we have provided a quick review of the work which is not included in Reference [3] below.

Mathematics 2019, 7, 1128; doi:10.3390/math7111128 www.mdpi.com/journal/mathematics

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Mathematics 2019, 7, 1128 2 of 14

Ayyapan et al. considered M/M/1 retrial queuing systems with loss and with DFB undernon-pre-emptive [4] and pre-emptive [5] priority service disciplines. Two types of customers arriveand the retrial, loss and feedback is allowed for low priority customers only. If the server is free atthe time of the arrival of low priority customer, then an incoming customer immediately begins to beserviced. After completion of service, the low priority customer either may re-join the orbit of infinitesize or leaves the system. If the server is busy then the low priority arriving customer either goes toorbit or becomes a source of repeated calls. If the server is free at the time of the arrival of high prioritycustomer, then the arriving call begins to be served immediately and high priority customer leavesthe system after service completion. In Reference [4] the non-pre-emptive priority service disciplineis used, i.e., if the server is engaging with a low priority customer and the higher priority customercomes then the high priority customer will get service only after completion of the service of the lowpriority customer who is in service. In Reference [5] the pre-emptive priority service is used, i.e., if theserver is engaging with low priority customer and the higher priority customer enters then the highpriority customer will get service immediately and the low priority customer who is in service goes toorbit without completion of his service. In both papers, matrix geometric method by Neuts [6] is used.

Belarbi et al. [7] focuses on an M/M/1/N < ∝ retrial queuing systems with single flow, loss and withtwo orbits for DFB. If upon arrival, the customers find the queue full, then they decide to stay for anattempt to get service or leave the system. Arriving customers who decide to stay for an attempt haveto choose one of the orbits in accordance with the probabilities that depend on thresholds in orbits.After the customer is served completely, they may decide to either attempt for re-service (feedback) orleave the system forever. In this case, the selection of one of the orbits occurs according to the Bernoullischeme. In Reference [7] the generating functions method is used.

Choi et al. [8] investigated an M/M/c/c retrial queues with geometric loss and with DFB whenc = 1, 2. An arriving customer is served immediately if one of the servers is idle, and if all servers arebusy, they will either join the retrial group to try again after a random amount of time or leave thesystem forever without service. On retrials, each customer is treated the same as a primary customer.If the retrial customer finds free servers, they receive service immediately, and if all the servers are busy,they will either leave the system forever or join the retrial group. All customers in the retrial groupact independently of each other. The retrial time is exponentially distributed with finite mean and isindependent of all previous retrial times and all other stochastic processes in the system. After thecustomer is served completely, they will decide either to join the retrial group again for another service(feedback) or to leave the system forever. They found the joint generating function of the number ofbusy servers and the number of customers in the retrial group by solving Kummer differential equationfor c = 1, and by the method of series solution for c = 1, 2.

Kumar et al. [9] considered an M/M/c/N + c queue with DFB and with constant retrial rate, whichis independent of the number of customers in the retrial group. Note that a customer in the orbit canenter the service facility at the retrial instant only when there are idle servers. The problem is solved bythe matrix-geometric method [6]. The same model was investigated in Reference [10] by Do where anefficient algorithm based on spectral expansion method by Mitrani and Chakka [11] is developed.

In Reference [12] Mokaddis et al. considered M/G/1 retrial DFB queue without a waiting spaceand single vacation where the server is subjected to starting failures. It is assumed that the retrial timeis governed by an arbitrary distribution and that only the customer at the head of the orbit queue isallowed access to the server. The server leaves for a vacation as soon as the system becomes empty.When the server returns from the vacation and finds no customers, it waits free for the first customerto arrive from outside the system. If the server is started successfully (with a certain probability), thecustomer gets service immediately. Otherwise, the repair for the server starts immediately and thecustomer must leave for the orbit and make a retrial later. The system size distribution at randompoints and various performance measures are derived.

It is important to note that in the works indicated above it is assumed that the departure andfeedback probabilities are constant quantities, i.e., they do not depend on the state of the system.

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This assumption significantly limits the application areas of the proposed models, since decisionsabout going into orbit or leaving orbit without getting re-service often depend on the current state ofthe system.

In the literature, models of queuing systems with DFB under variable input flow intensity havenot been studied. Based on this, in the present paper, we studied models of queuing systems withDFB in the presence of a Markov modulated Poisson process (MMPP) [13] in which the departure andfeedback probabilities depended on the current state of the system. Recently a similar model by usingthe asymptotic method was studied in Reference [14].

This paper is structured as follows. The mathematical model of the investigated system isdescribed in Section 2. The generator matrix of the three-dimensional Markov chain (3D MC) is createdin Section 3. Here the exact method to calculate the steady-state probabilities of the constructed 3DMC is indicated and explicit formulas for some performance measures of the investigated systemis developed as well. In Section 4, an approximate method for solving indicated above problems isproposed. Results of numerical experiments that show high accuracy of the approximate method aredemonstrated in Section 5. Conclusions are given in Section 6.

2. Mathematical Model

Pictorial representation of the investigated system is shown in Figure 1.

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It is important to note that in the works indicated above it is assumed that the departure and

feedback probabilities are constant quantities, i.e., they do not depend on the state of the system. This

assumption significantly limits the application areas of the proposed models, since decisions about

going into orbit or leaving orbit without getting re-service often depend on the current state of the

system.

In the literature, models of queuing systems with DFB under variable input flow intensity have

not been studied. Based on this, in the present paper, we studied models of queuing systems with

DFB in the presence of a Markov modulated Poisson process (MMPP) [13] in which the departure

and feedback probabilities depended on the current state of the system. Recently a similar model by

using the asymptotic method was studied in Reference [14].

This paper is structured as follows. The mathematical model of the investigated system is

described in Section 2. The generator matrix of the three-dimensional Markov chain (3D MC) is

created in Section 3. Here the exact method to calculate the steady-state probabilities of the

constructed 3D MC is indicated and explicit formulas for some performance measures of the

investigated system is developed as well. In Section 4, an approximate method for solving indicated

above problems is proposed. Results of numerical experiments that show high accuracy of the

approximate method are demonstrated in Section 5. Conclusions are given in Section 6.

2. Mathematical Model

Pictorial representation of the investigated system is shown in Figure 1.

Figure 1. Structure of the queuing system MMPP/M/K/K with delayed feedback.

The system contains 1K identical channels and incoming calls form the MMPP-flow. The

Markov chain (MC), which controls the intensities of the incoming flow, has a generating matrix (GM)

ij= of dimension NN , where ij determines the transition intensity from state i to

state j , Nji ...,,2,1, = , and =

−=N

ijj

ijii

,1

.

It is assumed that when control MC is in a state n , the intensity of the incoming stream is equal

Nnn ...,,2,1, = , and when the state of the control MC changes, the intensity of the incoming flow

Figure 1. Structure of the queuing system MMPP/M/K/K with delayed feedback.

The system contains K > 1 identical channels and incoming calls form the MMPP-flow. The Markovchain (MC), which controls the intensities of the incoming flow, has a generating matrix (GM) Σ = ‖σi j‖ofdimension N ×N, where σi j determines the transition intensity from state i to state j, i, j = 1, 2, . . . , N,

and σii = −N∑

j=1, j,iσi j.

It is assumed that when control MC is in a state n, the intensity of the incoming stream is equalλn, n = 1, 2, . . . , N, and when the state of the control MC changes, the intensity of the incoming flowalso changes instantly. The vector of intensities of the incoming flow is denoted byλ = (λ1, λ2, . . . , λN).

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Arriving from outside calls is called a primary call (p-call). An incoming call is accepted for serviceif at that moment there is a free channel; otherwise, i.e., if at that moment all channels are busy, then theincoming call is lost with a probability (w.p.) of one. After completion of the service process, the call,according to the Bernoulli scheme, either w.p. αk requires repeated service, or with a complementaryprobability leaves the system. The departure and feedback probabilities depend on a parameter k thatindicates the number of occupied channels immediately before the moment of a call’s departure, and itis assumed that αk > 0 for at least one k, k = 1, 2, . . . , K.

Feedback calls organize an orbit of repeated calls (r-calls) with a finite size R, 0 < R < ∞, i.e.,a feedback call is accepted into orbit if the total number of r-calls is less than R; otherwise, a feedbackcall leaves the system w.p. 1.

Service times for both types of calls are independent and identically distributed (i.i.d.) randomvariables (r.v.) and suppose that their distribution function is an exponential one with a commonaverage 1/µ.

Calls from orbit come through random times that obey the exponential distribution with the mean1/η. If upon arrival of an r-call all the channels of the system are busy, then it leaves the system w.p. βr,and with a complementary probability 1− βr returns to orbit, where r means the current number ofcalls in orbit, r = 1, 2, . . . , R. It is assumed that the random processes of changing the states of the MC,which control the intensity of the incoming flow, service of calls and the arriving of r-calls from orbit,are independent of each other.

The problem is to determine the stationary distribution of states of a given system. The solutionto this problem allows us to find the performance measures of the system under study: the averagenumber of busy channels (Kav), the average number of repeated calls in orbit (L0), as well as theprobability that all channels of the system are busy (CBP).

3. Method Based on Balance Equations

The state of the system at an arbitrary moment of the time is defined by the three-dimensionalvector (n, k, r), where n is the state of the MMPP-flow, k is total number of busy channels and r isnumber of retrial calls in orbit. Taking into account the type of distribution functions of all r.v. that areinvolved in the formation of the model we conclude that the state space of the investigated system isdetermined as follows:

E = {1, 2, . . . , N} × {0, 1, . . . , K} × {0, 1, . . . , R} . (1)

Transition intensity from state (n, k, r) to state (n′, k′, r′) is denoted by q((n, k, r), (n′, k′, r′)).These quantities are determined as follows:

• The transition (n, k, r)→ (n′, k, r), n′ , n is carried out with intensity σn,n′ when the state of theMMPP flow changes;

• The transition (n, k, r)→ (n, k + 1, r), k < K, is carried out with intensity λn upon receipt of thep-call;

• The transition (n, k, r)→ (n, k− 1, r), k > 0, r < R, is carried out with intensity kµ(1− αk) uponcompletion of the call service and having left the system;

• The transition (n, k, r)→ (n, k− 1, r + 1), k > 0, r < R, is carried out with intensity kµαk uponcompletion of the call service and its departure into orbit;

• The transition (n, k, R)→ (n, k− 1, R), k > 0, is carried out with intensity kµ upon completion ofthe call service;

• The transition (n, k, r)→ (n, k + 1, r− 1), k < K, r > 0, is carried out with intensity rηupon receiptof an r-call from orbit;

• The transition (n, K, r)→ (n, K, r− 1), r > 0, is carried out with intensity rηβr when r-call is lostfrom orbit.

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Therefore, the positive elements of GM of the investigated 3D MC are determined as follows:

q((n, k, r), (n′, k′, r′)) =

σnn′ , n′ , n, k′ = k, r′ = r,λn, n′ = n, k′ = k + 1, r′ = r,kµ(1− αk), k > 0, r < R, n′ = n, k′ = k− 1, r′ = r,kµαk, k > 0, r < R, n′ = n, k′ = k− 1, r′ = r + 1,kµ, k > 0, r = R, n′ = n, k′ = k− 1, r′ = r,rη, k < K, r > 0, n′ = n, k′ = k + 1, r′ = r− 1,rηβr, k = K, r > 0, n′ = n, k′ = k, r′ = r− 1 .

(2)

From relations in Equation (2) we conclude that the investigated 3D MC with state space inEquation (1) is irreducible, i.e., in this chain there exist steady-state probabilities. Let p(n, k, r) denotesteady-state probability of state (n, k, r) ∈ E. These probabilities satisfy the system of equilibriumequations (SEE), which are compiled using relations in Equation (2). The explicit form of this SEE isnot given here due to the obviousness of its derivation and cumbersome nature.

Finding the steady-state probabilities allows us to calculate the performance measures of thesystem under study. So, the average number of busy channels and the average number of r-calls inorbit are determined as the mathematical expectations of the corresponding r.v., i.e.,

Kav =N∑

n=1

K∑k=1

kR∑

r=0

p(n, k, r); (3)

Lo =N∑

n=1

R∑r=1

rK∑

k=0

p(n, k, r). (4)

The probability that all channels of the system are busy is determined as follows:

CBP =∑

(n,K,r)∈Ep(n, K, r). (5)

Note that, due to the complex structure of the obtained GM (see Equation (2)), it is not possible tofind an analytical solution to the corresponding SEE for steady-state probabilities. Therefore, to solveit, one has to use numerical methods of linear algebra, and modern software packages allow us tosolve SEE of moderate dimension (see References [15,16]). However, for models of large dimensions,these methods will require a huge runtime, and often suffer from computational instability. Based onthis, an alternative (approximate) method for solving this problem is given below. It is based on thehierarchical space merging approach for multidimensional MCs [17].

4. Method Based on Space Merging Hierarchy

For the correct application of the developed method, we assume that the MMPP flow is inertial.This means that transitions between its states occur with low intensities. When fulfilling this assumption,we consider the following splitting of the state space from Equation (1):

E =N∪

n=1En , En∩En′ = ∅, n , n′, (6)

where En ={(n, k, r) ∈ E : k = 0, 1, . . . , K; r = 0, 1, . . . , R}, n = 1, 2, . . . , N.

Based on splitting Equation (6) in the state space of Equation (1), the following merge function isdefined:

U1(n, k, r) =< n >, (n, k, r) ∈ En , (7)

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where < n > is the merged state which includes all states from the class En, n = 1, 2, . . . , N. We denoteΩ1 = {< n > : n = 1, 2, . . . , N} .

The steady-state probabilities of the initial 3D MC are determined as follows (see [17]):

p(n, k, r) ≈ ρn(k, r)π1(< n >) , (8)

where ρn(k, r) is the probability of state (k, r) within splitting model with state space En, π1(< n >) isthe probability of merged state < n >∈ Ω 1.

From Equation (8) we conclude that to calculate the steady-state probabilities of the initial 3D MC,it is necessary to find the state probabilities of the 2D MCs (their number is equal N) and one 1D MC.

The transitions between the states of the Markov chain, which control the intensity of the incomingflow, do not depend on the status of channels (busy or free) and on the intensity of r-calls. Therefore,its stationary distribution π1(< n >), < n >∈ Ω1 is uniquely determined using its generating matrix Σ.In other words, to solve the problem, it is enough to find the stationary distribution of 2D MCs withstate spaces En, n = 1, 2, . . . , N.

A fragment of the transition graph between the states of a split model with a state space En isshown in Figure 2.

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, ,,1

nnEEEE nn

N

n

n==

=

(6)

where ( ) ....,,2,1,...,,1,0;...,,1,0:,, NnRrKkErknEn ==== Based on splitting Equation (6) in the state space of Equation (1), the following merge function

is defined:

( ) ( ) ,,,,,,1 nErknnrknU = (7)

where n is the merged state which includes all states from the class nE , Nn ...,,2,1= . We

denote ....,,2,1:1 Nnn == The steady-state probabilities of the initial 3D MC are determined as follows (see [17]):

( ) ( ) ( ),,,, 1 nrkrknp n (8)

where ( )rkn , is the probability of state ),( rk within splitting model with state space nE ,

( ) n1 is the probability of merged state .1 n From Equation (8) we conclude that to calculate the steady-state probabilities of the initial 3D

MC, it is necessary to find the state probabilities of the 2D MCs (their number is equal N ) and one 1D MC.

The transitions between the states of the Markov chain, which control the intensity of the

incoming flow, do not depend on the status of channels (busy or free) and on the intensity of r-calls.

Therefore, its stationary distribution ( ) 11 , nn is uniquely determined using its generating matrix . In other words, to solve the problem, it is enough to find the stationary

distribution of 2D MCs with state spaces ....,,2,1, NnEn =

A fragment of the transition graph between the states of a split model with a state space nE is

shown in Figure 2.

Figure 2. Fragment of the transition graph for the splitting model with state space En.

Computational difficulties arise in solving the latter problem if the number of channels and the

size of the orbit are huge. Therefore, the merge procedure (second level of the hierarchy) is applied

to these 2D MCs. Since all splitting models are identical, we fix the value of parameter n and

consider the split model with the state space .nE

For correct application of the proposed method, it is assumed that

= Nnn ...,,2,1:min . Under this assumption consider following splitting in class nE :

,,,0

rrEEEE rnr

n

R

r

r

nn==

=

(9)

where ( ) ....,,1,0,...,,1,0:, RrKkErkE nr

n ===

Figure 2. Fragment of the transition graph for the splitting model with state space En.

Computational difficulties arise in solving the latter problem if the number of channels and thesize of the orbit are huge. Therefore, the merge procedure (second level of the hierarchy) is applied tothese 2D MCs. Since all splitting models are identical, we fix the value of parameter n and consider thesplit model with the state space En.

For correct application of the proposed method, it is assumed that min{λn : n = 1, 2, . . . , N} >> η.Under this assumption consider following splitting in class En:

En =R∪

r=0Ern , E

rn∩Er

′n = ∅, r , r′, (9)

where Ern ={(k, r) ∈ En : k = 0, 1, . . . , K

}, r = 0, 1, . . . , R.

As above (see Equation (7)), based on splitting Equation (9) in the state space En the followingmerge function is defined:

U2((k, r)) =< r > , (k, r) ∈ Ern,

where < r > is the merge state which includes all states from the class Ern. We denote Ω2 ={< r > : r = 0, 1, . . . , R}.

In accordance [17] we have:ρn(k, r) ≈ ρrn(k)πn2(< r >) , (10)

where ρrn(k) is the probability of state (k, r) within the splitting model with space Ern, πn2(< r >) is theprobability of the merged state < r >∈ Ω2 .

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The transition graph between the states of a split model with a state space Ern, r = 0, 1, . . . , R− 1,is shown in Figure 3.

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As above (see Equation (7)), based on splitting Equation (9) in the state space nE the following

merge function is defined:

( )( ) ,,2 = rrkU ( ) ,,r

nErk

where r is the merge state which includes all states from the class rnE . We denote

....,,1,0:2 Rrr == In accordance [17] we have:

( ) ( ) ( ),, 2 rkrknr

nn (10)

where ( )krn is the probability of state ),( rk within the splitting model with space r

nE ,

( ) rn2 is the probability of the merged state .2 r The transition graph between the states of a split model with a state space ,1...,,1,0, −= RrE rn

is shown in Figure 3.

Figure 3. Graph of the splitting model with state space .1...,,1,0, −= RrE rn

In a class ,...,,1,0, RrE rn = in a state vector, the second component is constant and equal r .

Therefore, in a splitting model with space r

nE each state ),( rk can be specified only by the first

component, i.e., hereinafter state r

nErk ),( referred to as Kkk ...,,1,0, = . Then from relations

in Equation (2) we conclude that the intensities of transitions between the states of the splitting model

with space ,1...,,1,0, −= RrE rn are independent of the parameter r , and are determined as

follows (see Figure 3):

( )( )

−=−

+==

.1 if,1

,1 if,,

kkk

kkkkq

k

n

n

(11)

From Equation (11) we conclude that the probabilities of the states of splitting models with space

,1...,,1,0, −= RrE rn coincide with the state probabilities of the one-dimensional birth–death

process in which the birth intensity is equal to n and the death intensity in the state k is equal to

( ) ....,,1,1 Kkk k =− For a splitting model with a space R

nE , the intensities of transitions

between its states are determined similarly to Reference (11), but in this model, on the right side of

the indicated formulae (in the second line), the term k−1 is replaced by 1. This is because if the

call arrives at the moment when the orbit is completely full then it is lost w.p. 1. Therefore, the state

probabilities of the splitting model with space R

nE coincide with the probabilities of the Erlang’s

M/M/K/K system with a load n . After certain mathematical transformations, we obtain that the intensities of transitions between

states of space 2 are determined as follows:

( )( )

−=

+==

,1 if,

,1 if,,

rrnr

rrrrq

r

n

n (12)

Figure 3. Graph of the splitting model with state space Ern, r = 0, 1, . . . , R− 1.

In a class Ern, r = 0, 1, . . . , R, in a state vector, the second component is constant and equalr. Therefore, in a splitting model with space Ern each state (k, r) can be specified only by the firstcomponent, i.e., hereinafter state (k, r) ∈ Ern referred to as k, k = 0, 1, . . . , K. Then from relations inEquation (2) we conclude that the intensities of transitions between the states of the splitting modelwith space Ern, r = 0, 1, . . . , R− 1, are independent of the parameter r, and are determined as follows(see Figure 3):

qn(k, k′) ={λn, if k′ = k + 1 ,kµ(1− αk), if k′ = k− 1 .

(11)

From Equation (11) we conclude that the probabilities of the states of splitting models withspace Ern, r = 0, 1, . . . , R− 1, coincide with the state probabilities of the one-dimensional birth–deathprocess in which the birth intensity is equal to λn and the death intensity in the state k is equal tokµ(1− αk) , k = 1, . . . , K . For a splitting model with a space ERn , the intensities of transitions betweenits states are determined similarly to Reference (11), but in this model, on the right side of the indicatedformulae (in the second line), the term 1− αk is replaced by 1. This is because if the call arrives at themoment when the orbit is completely full then it is lost w.p. 1. Therefore, the state probabilities of thesplitting model with space ERn coincide with the probabilities of the Erlang’s M/M/K/K system with aload νn .

After certain mathematical transformations, we obtain that the intensities of transitions betweenstates of space Ω2 are determined as follows:

qn(< r >,< r′ >) ={

Λn, if r′ = r + 1 ,rΨr(n), if r′ = r− 1 ,

(12)

where Λn = µK∑

k=1kαkρrn(k) , Ψr(n) = η(1− (1− βr)ρrn(K)) .

Therefore, from relations in Equation (12) we conclude that the probabilities of merged states arecalculated as follows:

πn2(< r >) =Λrn

r!r∏

i=1Ψi(n)

πn2(< 0 >), r = 1, . . . , R, (13)

where πn2(< 0 >) is found from normalizing condition, i.e.,R∑

r=0πn2(< r >) = 1.

Finally, taking into account relations from Equations (8), (10), (11) and (13), approximate values ofthe steady-state of the initial 3D MC are found. Further, after certain mathematical transformations,we obtain the following approximate formulas for calculating the performance measures of the systemunder study:

Kav ≈N∑

n=1

π1(< n >)K∑

k=1

kR∑

r=0

ρrn(k)πn2(< r >); (14)

Lo ≈N∑

n=1

π1(< n >)R∑

r=1

rπn2(< r >); (15)

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CBP ≈N∑

n=1

π1(< n >)R∑

r=0

ρrn(K)πn2(< r >). (16)

Special cases. Note that in the cases αk = α for all k = 1, 2, . . . , K, and βr = β for all r = 1, 2, . . . , R,the Equations (11–16) are even more simplified. Moreover, in this case it is possible to obtain explicitsolutions for models with an infinite number of channels and/or an infinite orbit size.

So, for a model with a finite number of channels (K < ∞) and an infinite orbit size (R = ∞), thestate probabilities inside all splitting models with space Ern, r = 0, 1, 2, . . . , coincide with the stateprobabilities of the Erlang model M/M/K/K with load νn/(1− α). The probabilities of merged statescoincide with the probabilities of the states of the model M/M/∞with the load θ̃n = Λ̃n/Ψ̃(n), i.e.,

πn2(< r >) =θ̃rnr!

exp(−θ̃n

), r = 0, 1, 2, . . . ,

where Λ̃n = µαBav(νn

1−α , K)

, Ψ̃(n) = η(1− (1− β)EB

(νn

1−α , K))

.Hereinafter, EB(x, K) and Bav(x, K) denote the Erlang B-formula and the average number of busy

channels for the M/M/K/K system with a load x, respectively, i.e.,

EB(x, K) =xK

K!/

K∑k=0

xk

k!; Bav(x, K) = x(1− EB(x, K)) .

Then from Equations (14)–(16) we obtain that the performance measures of this model aredetermined as follows:

Kav ≈N∑

n=1

π1(< n >)Bav(νn

1− α , K);

Lo ≈N∑

n=1

π1(< n >) θ̃n;

CBP ≈N∑

n=1

π1(< n >)EB(νn

1− α , K).

In a model with an infinite number of channels (K = ∞) and an infinite orbit size (R = ∞), the stateprobabilities inside all split models with space Ern, r = 0, 1, 2, . . . , coincide with the state probabilities ofthe model M/M/∞ with a load νn/(1− α); the probabilities of merged states πn2(< r >) , r = 0, 1, 2, . . .,coincide with the probabilities of the states of the model M/M/∞ with a load θ̂n = Λ̂n/η, whereΛ̂n = µνn α1−α . In this model, we have CBP = 0, however the remaining performance measures aredefined as follows:

Kav ≈1

1− α

N∑n=1

π1(< n >) νn; Lo ≈N∑

n=1

π1(< n >) θ̂n.

5. Numerical Results

Here we present the results of numerical experiments. These experiments had two goals: (1)investigation of the accuracy of the developed approximate method; (2) investigation of the behaviorof the performance measures versus both the feedback αk, k = 1, . . . , K and retrial probabilitiesβr, r = 1, . . . , R.

Exact values of the steady-state probabilities were calculated from the corresponding SEE andthe accuracy of their calculation might be estimated using various similarity norms. For this purpose,the cosine similarity norm was used, since in our experiments it estimated not only the orientation ofthe steady-state probability vectors in various (exact and approximate) approaches, as well as their

Mathematics 2019, 7, 1128 9 of 14

Euclidean proximity. This is because, according to the normalization condition, the ends of thesevectors were in the same hyper plane of the Euclidean space of dimension N(K + 1)(R + 1).

For the sake of clarity, let us assume that the MMPP flow is determined using a MC with threestates (i.e., N = 3) and the following generating matrix:

Σ = ‖−35 15 2015 −25 1010 10 −20

‖.

The vector of intensities of the incoming flow is defined as follows: λ = (20, 30, 40); rateof feedback calls from orbit is η = 15. It is assumed that the parameters αk, k = 1, . . . , K, andβr, r = 1, . . . , R, are constant, i.e., αk = α = 0.2, βr = β = 0.9.

The results of the corresponding numerical experiments are shown in Table 1. From this tablewe concluded that the steady-state probabilities were calculated with high accuracy, i.e., similaritynorm is close to one. It is important to note that the values of the performance measures of Equations(3) and (5) in both approaches are very close to each other and small deviations are observed only incalculation of the performance measure of Equation (4), and such errors for engineering calculations arequite acceptable. Note that the indicated deviations are expected, since the performance measures ofEquations (3) and (4) are linear forms with respect to the steady-state probabilities, and their coefficientsare sufficiently large numbers for models of large dimension.

Table 1. Results of numerical experiments, EV—exact values, AV—approximate values.

µ (K, R)Cosine

SimilarityCBP Kav Lo

EV AV EV AV EV AV

3

(5, 5) 0.9163 0.6179 0.6366 4.4708 4.4891 0.1013 0.0274(5, 7) 0.9164 0.6179 0.6366 4.4708 4.4891 0.1013 0.0274

(5, 10) 0.9164 0.6179 0.6366 4.4708 4.4891 0.1013 0.0274(7, 5) 0.8901 0.4865 0.5045 6.1453 6.1449 0.1117 0.0359(7, 7) 0.8901 0.4866 0.5045 6.1453 6.1449 0.1117 0.0359

(7, 10) 0.8901 0.4866 0.5045 6.1453 6.1449 0.1117 0.0359(10, 5) 0.8476 0.3144 0.3318 8.4491 8.3697 0.1264 0.0452(10, 7) 0.8476 0.3144 0.3318 8.4491 8.3697 0.1264 0.0452

(10, 10) 0.8476 0.3144 0.3318 8.4491 8.3697 0.1264 0.0452

5

(5, 5) 0.8878 0.4372 0.4567 4.0643 4.0799 0.1469 0.0380(5, 7) 0.8878 0.4372 0.4567 4.0643 4.0799 0.1469 0.0380

(5, 10) 0.8878 0.4372 0.4567 4.0643 4.0799 0.1469 0.0380(7, 5) 0.8552 0.2745 0.2929 5.3877 5.3604 0.1691 0.0460(7, 7) 0.8552 0.2746 0.2929 5.3877 5.3604 0.1691 0.0460

(7, 10) 0.8552 0.2746 0.2929 5.3877 5.3604 0.1691 0.0456(10, 5) 0.8215 0.1069 0.1269 6.8321 6.7419 0.1951 0.0508(10, 7) 0.8215 0.1069 0.1269 6.8321 6.7419 0.1951 0.0508

(10, 10) 0.8215 0.1069 0.1269 6.8321 6.7419 0.1951 0.0508

7

(5, 5) 0.8718 0.3060 0.3256 3.6551 3.6767 0.2086 0.0441(5, 7) 0.8718 0.3060 0.3256 3.6551 3.6767 0.2086 0.0441

(5, 10) 0.8718 0.3060 0.3256 3.6551 3.6767 0.2086 0.0441(7, 5) 0.8482 0.1479 0.1657 4.6165 4.6017 0.2109 0.0496(7, 7) 0.8482 0.1479 0.1657 4.6165 4.6017 0.2109 0.0496

(7, 10) 0.8482 0.1479 0.1657 4.6164 4.6017 0.2109 0.0496(10, 5) 0.8378 0.0320 0.0445 5.3628 5.3598 0.2210 0.0515(10, 7) 0.8378 0.0312 0.0445 5.3628 5.3598 0.2210 0.0515

(10, 10) 0.8378 0.0312 0.0445 5.3628 5.3598 0.2210 0.0515

Mathematics 2019, 7, 1128 10 of 14

Next we considered the second goal, i.e., we studied the behavior of the performance measuresof the system versus the feedback and retrial probabilities. First, we considered the systems withstate-independent feedback and retrial probabilities. The corresponding results for the initial dataindicated above and for R = 10 are shown in Figure 4. It can be seen from the graphs that a simultaneousincrease in the feedback probability and a decrease in the probability of loss from orbit when allchannels were occupied led to an increase in all performance measures of the system. With an increasein the number of channels, the functions Kav and Lo grew with high speed (see Figure 4a,b), whilethe values of the function CBP approached each other for the selected pairs of parameters (α, β) (seeFigure 4c).

Next, we considered the behavior of the performance measures from Equations (3) and (5) for thesystems with state-dependent feedback and retrial probabilities.

The results for the system with state-dependent feedback probabilities and constant retrialprobabilities for R = 5 are shown in Figure 5. Here we considered two schemas for the values offeedback probabilities. In the first scheme parameters αk increased with respect to k, i.e., αk = k/(k + 1),while in the second one αk decreased with respect to k, i.e., αk = 1/(k + 1) , k = 1, . . . , K. It can be seenfrom the graphs that in both schemas values of performance measures Kav (see Figure 5a) and CBP(see Figure 5c) were almost the same. However, values of performance measure Lo (see Figure 5b)depended on the scheme of changing of feedback probabilities. For any number of channels, increasingof feedback probabilities led to an increase in the number of calls in orbit. This fact was an expectedone since with the increase of the feedback probabilities, the intensity of calls into orbit increases.From Figure 5b it can be seen that in the first scheme for changing the feedback probabilities, thenumber of calls in the orbit took its maximum value at K = 3, and then it decreased. This result at firstglance seemed unexpected. In this regard, we noted that unlike the case of state-independent feedbackprobabilities (see Figure 4b) here, behavior of function Lo essentially depended on the distributionof the number of busy channels. In addition, distribution of the number of busy channels in turndepended on all structural and load parameters of system. Therefore, it was impossible to predictbehavior of this function in the case of state-dependent feedback probabilities.

The results for the system with state-dependent retrial probabilities and constant feedbackprobabilities for K = 5 are shown in Figure 6. Here we also consider two schemas for the values ofretrial probabilities. In the first scheme parameters βr increased with respect to r, i.e., βr = r/(r + 1),while in the second one βr are decreased with respect to r, i.e., βr = 1/(r + 1) , r = 1, . . . , R. Unlike theprevious figures here all performance measures were increasing ones and their values were almostindependent of the retrial probabilities variation schemes. Note that from these graphs we concludedthat for large restrictions on the orbit size, the values of all performance measures turned out to beslightly smaller in the first scheme. These facts were expected, since with an increase in the probabilityof leaving the system, the number of calls in orbit decreased and therefore the average number of busychannels as well as the probability that all channels were busy decreased.

Mathematics 2019, 7, 1128 11 of 14

Mathematics 2019, 7, x 11 of 15

increase in the number of calls in orbit. This fact was an expected one since with the increase of the

feedback probabilities, the intensity of calls into orbit increases. From Figure 5b it can be seen that in

the first scheme for changing the feedback probabilities, the number of calls in the orbit took its

maximum value at 3=K , and then it decreased. This result at first glance seemed unexpected. In this regard, we noted that unlike the case of state-independent feedback probabilities (see Figure 4b)

here, behavior of function oL essentially depended on the distribution of the number of busy

channels. In addition, distribution of the number of busy channels in turn depended on all structural

and load parameters of system. Therefore, it was impossible to predict behavior of this function in

the case of state-dependent feedback probabilities.

(a)

(b)

.

(c)

Figure 4. Dependency avK (a), oL (b) and CBP (c) on K for state-independent feedback and retrial

probabilities.

Figure 4. Dependency Kav (a), Lo (b) and CBP (c) on K for state-independent feedback andretrial probabilities.

Mathematics 2019, 7, 1128 12 of 14Mathematics 2019, 7, x 12 of 15

(a)

(b)

(c)

Figure 5. Dependency avK (a), oL (b) and CBP (c) on K for state-dependent feedback

probabilities.

The results for the system with state-dependent retrial probabilities and constant feedback

probabilities for 5=K are shown in Figure 6. Here we also consider two schemas for the values of

retrial probabilities. In the first scheme parameters r increased with respect to r , i.e.,

( )1+= rrr , while in the second one r are decreased with respect to r , i.e., ( ) ....,,1,11 Rrrr =+= Unlike the previous figures here all performance measures were

Figure 5. Dependency Kav (a), Lo (b) and CBP (c) on K for state-dependent feedback probabilities.

Mathematics 2019, 7, 1128 13 of 14

Mathematics 2019, 7, x 13 of 15

increasing ones and their values were almost independent of the retrial probabilities variation

schemes. Note that from these graphs we concluded that for large restrictions on the orbit size, the

values of all performance measures turned out to be slightly smaller in the first scheme. These facts

were expected, since with an increase in the probability of leaving the system, the number of calls in

orbit decreased and therefore the average number of busy channels as well as the probability that all

channels were busy decreased.

(a)

(b)

(c)

Figure 6. Dependency avK (a), oL (b) and CBP (c) on R for state-dependent retrial probabilities.

Figure 6. Dependency Kav (a), Lo (b) and CBP (c) on R for state-dependent retrial probabilities.

6. Conclusions

The paper proposes mathematical models of multichannel systems with an MMPP flow anddelayed feedback, in which the feedback is carried out by returning part of the initial calls for receivingrepeated service. The probabilities of feedback or departure depend on the current number of busychannels in the system. In addition, the probabilities of reneging of calls from orbit without receivingre-service or returning to orbit for re-circulation (if all the system channels are busy upon their arrival)depend on the number of calls in orbit. We studied models with finite orbit sizes for feedback calls anddeveloped exact and approximate methods for calculating their characteristics.

Mathematics 2019, 7, 1128 14 of 14

As directions for further research, the study of similar models in the presence of a MarkovArrival Process (MAP) flow of initial calls and phase-type (PH) distribution for the service time ofheterogeneous calls is proposed. Of particular interest are also the problems of optimizing systemswith feedback regarding the selected criteria for the quality of their functioning. These problems arethe subject of special studies.

Author Contributions: Conceptualization, A.M. and S.A.; methodology, A.M. and S.A.; software, S.A.; validation,S.A.; formal analysis, A.M., S.A. and J.S.; investigation, J.S., A.M.; writing—original draft preparation, A.M. andJ.S.; writing—review and editing, A.M. and J.S.; supervision A.M.; project administration S.A.

Funding: The work of J.S. was supported by the EFOP-3.6.1-16-2016-00022 project. The project was co-financedby the European Union and the European Social Fund.

Acknowledgments: The authors are very grateful to the reviewers for their valuable comments and suggestionsthat improved the quality and the presentation of the paper.

Conflicts of Interest: The authors declare no conflict of interest.

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http://dx.doi.org/10.1002/j.1538-7305.1963.tb00510.xhttp://dx.doi.org/10.1051/ro/1977110403451http://dx.doi.org/10.5120/672-945http://dx.doi.org/10.2478/ausm-2014-0009http://dx.doi.org/10.1016/S0898-1221(98)00160-6http://dx.doi.org/10.1016/j.apm.2008.05.011http://dx.doi.org/10.1016/j.apm.2009.10.025http://dx.doi.org/10.1016/0166-5316(94)00025-Fhttp://dx.doi.org/10.1016/0166-5316(93)90035-Shttp://dx.doi.org/10.1134/S0005117910070118http://creativecommons.org/http://creativecommons.org/licenses/by/4.0/.

Introduction Mathematical Model Method Based on Balance Equations Method Based on Space Merging Hierarchy Numerical Results Conclusions References